Fuzzy rule base design using tabu search algorithm for nonlinear system modeling

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ISA Transactions 4urinfo2u seveturule bases are compared with those belonging to other modeling approaches in the literature. The simulation results indicate that the method basedon the use of a TSA performs an important and very effective modeling procedure in fuzzy rule base design in the modeling of the nonlinear orcomplex systems.c 2007, ISA. Published by Elsevier Ltd. All rights reserved.Keywords: Fuzzy systems; Fuzzy modeling; Fuzzy rule base design; Nonlinear system modeling; Tabu search1. IntroductionModeling is an essential part of control design as it providesa way of formally proving properties of the closed loop system.There are many reasons why a model of a process may berequired. It could, for example, be used to simulate the realprocess or it could be used to design a controller. If a goodmodel of the system can be obtained by using knowledge andobservations, accurate and efficient solutions to engineeringproblems such as system design and simulation, processanalysis, reliability of the measured data, optimization of thesystems can be provided. Conventional system theory relieson crisp mathematical models of systems, such as algebraicand differential equations. However, many real world systemsare inherently nonlinear and cannot be represented by linearmodels used in conventional system identification. For a largenumber of practical problems, the gathering of an acceptablea fast and efficient solution to complex or nonlinear systemmodeling problems. One of the most successful expert systemmethods is fuzzy logic based modeling approach [1].Rule based fuzzy modeling is a powerful technique forthe modeling of partly known nonlinear systems. In themodeling of complicated and/or ill-defined processes, precisemathematical models may fail to give satisfactory results. Insuch cases, fuzzy models can effectively integrate informationfrom different sources (such as physical laws, empirical models,and experimental measurements) and they may be used toproperly depict the system uncertainty. If fuzzy rule basedmodels can be identified, fuzzy controllers can be obtained fromthe fuzzy models and genuine control engineering issues canbe addressed for fuzzy controllers, namely stability, robustnessetc. [2].In designing fuzzy models, a major difficulty in theFuzzy rule base design using tabsystem mAytekinDepartment of Electrical Electronic EngineeReceived 15 February 2007; received in revisedAvailable onlineAbstractThis paper presents an approach to fuzzy rule base design using tabevolve the structure and the parameter of fuzzy rule base. The use of thdetermination of fuzzy rule base parameters, leads to a significant improof the presented method, several numerical examples given in the literadegree of knowledge needed for physical modeling is a difficult,time consuming and expensive or even impossible task. Expertsystem techniques based on artificial intelligence can provide Tel.: +90 (0.352) 437 49 37x32255; fax: +90 (0.352) 437 57 84.E-mail address: bagis@erciyes.edu.tr.0019-0578/$ - see front matter c 2007, ISA. Published by Elsevier Ltd. All rightsdoi:10.1016/j.isatra.2007.09.0017 (2008) 3244www.elsevier.com/locate/isatranssearch algorithm for nonlinearodelingBagisg, Erciyes University, 38039 Kayseri, Turkeyrm 29 August 2007; accepted 4 September 20072 October 2007earch algorithm (TSA) for nonlinear system modeling. TSA is used toTSA, in conjunction with a systematic neighbourhood structure for thement in the performance of the model. To demonstrate the effectivenessre are examined. The results obtained by means of the identified fuzzyidentification of an optimized fuzzy rule base is encountered.Traditional fuzzy rule design is based upon a human operatorsexperienced knowledge. Not only is this design method timeconsuming since it uses trial and error to find good fuzzyrules and membership functions, it is also not guaranteedto find optimal or near optimal fuzzy rules and membershipfunctions [3,4].reserved.tioA. Bagis / ISA TransacThere are many studies on fuzzy modeling in the literature.Takagi and Sugeno proposed a new type of fuzzy modelcalled TakagiSugeno fuzzy model (TS fuzzy model) [5].Furthermore, this study proposes a procedure to identify theTS fuzzy model from inputoutput data of systems. Sugenoand Yasukawa discussed a general approach to qualitativemodeling based on fuzzy logic [6]. It proposes the use of afuzzy clustering method to identify the structure of a fuzzymodel. Wang and Lee proposed a method based on theback-propagation algorithm to determine significant variableswithout the need of human inspection [7]. A novel fuzzymodeling algorithm was presented by Kim et al. [8]. In thestudy, fuzzy C regression model clustering algorithm and agradient descent algorithm are used. Dynamic versions offuzzy logic systems and their nonsingleton generalizationswere investigated by Mouzouris and Mendel [9]. They useda dynamic learning algorithm to train the system parameters.Wang and Langari presented a fuzzy logic approach tocomplex system modeling that is based on fuzzy discretizationtechnique [10]. A great advantage of the presented approachwas that it could determine the premise structure of a modeloff-line. In [11], a method of TakagiSugenoKang fuzzymodel generation using numerical data as a starting point wasdescribed by Kukolj.In addition, many other methods such as neural networksbased methods were suggested and were used to build fuzzymodels [1215]. Several models for the identification andcontrol of nonlinear dynamic systems were suggested byNarendra and Parthasarathy [13]. These models, which includemultilayer neural networks as well as linear dynamics, can beviewed as generalized neural networks. For the adjustment ofparameters, static and dynamic back-propagation methods areused. Jang described an adaptive network based fuzzy inferencesystem [12]. The presented method employed a hybrid learningprocedure to construct an inputoutput mapping based onboth human knowledge and stipulated inputoutput datapairs. By using the architecture of the fuzzy-neural network,simple and effective fuzzy rule based models of complexsystems from inputoutput data were developed by Lin andCunningham [15]. Babuska and Verbruggen presented anoverview of neuro-fuzzy modeling methods for nonlinearsystem identification [14].Different optimization algorithms were also used forfuzzy system modeling [1623]. Hanss presented a specialfuzzy modeling method for developing multivariable fuzzymodels [20]. This fuzzy model identification procedure wascarried out by applying a special clustering method. Kilicet al. proposed a fuzzy system modeling approach based ona probabilistic hill-climbing algorithm as a data analysis andapproximate reasoning tool [19]. Evsukoff et al. used a leastsquares error minimization method for nonlinear fuzzy modelidentification [16]. A genetic based neuro-fuzzy approach formodeling and control of dynamical systems was presented byFarag et al. [23]. The proposed approach combined the merits ofthe fuzzy logic theory, neural networks, and genetic algorithms.Wu and Yu used a genetic algorithm and gradient descent basedapproach to optimize the structure of a TakagiSugenoKangns 47 (2008) 3244 33model [17]. An evolutionary design method of fuzzy rule basefor nonlinear system modeling and control was presented byKang et al. [22]. Cordon et al. presented an overview of therepresentative methods in genetic fuzzy systems proposed inthe literature [21].This paper presents an approach to fuzzy rule base designusing a tabu search algorithm (TSA) for nonlinear systemmodeling. TSA is used to evolve the structure and theparameters of fuzzy rule base. To check the effectiveness of thepresented approach, several numerical examples given in theliterature are examined. The results obtained by means of theidentified fuzzy rule bases are compared with those belongingto other modeling approaches in the literature. The remainderof the paper is organized as follows. In Section 2, a brief reviewof the TSA is introduced. Structure of the identified fuzzy rulebase and design method is developed in Section 3. Simulationresults and some conclusions are given in Sections 4 and 5,respectively.2. Tabu search algorithm (TSA)TSA was proposed by Glover as an intelligent optimizationtechnique to overcome local optimality in solution processes forhard combinatorial optimization problems [24]. This algorithmconsists of the systematic prohibition of some solutions toprevent cycling and to avoid the risk of trapping at a localoptimum. New solutions are searched in the neighbourhood ofthe current one. In TSA, there are several important concepts:Neighbourhood: To improve the quality of the solutions visited,the search moves from one solution to another using aneighbourhood structure. The neighbourhood of a solution isthe set of all formations that can be arrived at by a move.The move is a process that transforms the current solution toits neighbouring solution. If, after making a move, a solutionis found which is better than all solutions found in priormoves than the new solution is saved as the new best solution.This neighbourhood procedure is applied to the only one bestsolution of each generation. Considering a chromosome with nbits, the number of the neighbour solutions is n, as depicted inFig. 1. To obtain the neighbour solutions for the best solution,encoding and decoding operations are applied. In this work,using different bit sizes for each of the parameters we have triedto achieve a good balance between the algorithm performanceand the computational cost, and we finally adopt a bit sizeof 8. Therefore, total number of bits required for defining afuzzy rule base parameter set is [rule number parameternumber bit number]. According to simulation results, this bitsize is suitable for the main process in the algorithm describedin this paper.Tabu list: One of the main ideas of TSA is the use of aflexible memory (tabu list). The objective of the tabu list is toexclude moves which would bring the algorithm back to whereit was at some previous iteration and keep it trapped in a localminimum. To avoid performing a move returning to a recentlyvisited region, the reverses of last moves are forbidden. In otherwords, these moves are considered as tabu and are recordedin the tabu list. The tabu list stores all tabu moves that areact(a) input and output variables34 A. Bagis / ISA TransFig. 1. Neighbourhood structure for a solution.Table 1The values of the control parameters for TSATabu search algorithmRecency factor (r ) 0.2Frequency factor ( f ) 2.0Generation number 2000Tabu conditions (1) Recency (k) > (r M),(2) Frequency (k) < ( f avgfreq)Number of variables for a rule 9, for system with 2 inputs and 1 output13, for system with 3 inputs and 1 outputNumber of neighbourhood pervariable8Tabu list size Rule number number of the variablesnot permitted to be applied to the present solution. This list isinitialized empty, constructed in consecutive iterations of thesearch and updated circularly in later iterations.Tabu conditions: The number of the possible moves isdetermined by the tabu list. If a solution produced by a movethat is not on the tabu list is better than all solutions found inprior iterations, then this solution is saved as the best solution.At each of the iterations in the optimization process, candidatesolutions are checked with respect to tabu conditions, andhence, the next solution is determined depending on evaluationvalues and tabu conditions. The re-generation of a solutionpreviously obtained is avoided by using the tabu conditions(or restrictions) on the possible moves. The tabu conditionsare usually based on two important factors: frequency memoryand recency memory. Frequency memory keeps the knowledgeof how often the same solutions have been made in the past.Recency memory prevents cycles of length less than or equal toa predetermined number of iterations. The tabu conditions usedin this work are as the following: If the element k of a solutionvector does not satisfy the conditions (i) recency (k) > r.M ,(ii) frequency (k) < f.avgfreq, then it is accepted as tabu andnot used to create a neighbour. Here, r and f are recency andfrequency factors, M is the number of elements in the solutionvector, and avgfreq is the average frequency value. In this work,after some experimentation, the algorithm parameters r and fhave been set to the values of 0.2 and 2.0, respectively. Thecontrol parameter values for the TS algorithm are given inTable 1. For all of the simulations, the iteration or generationnumber is taken as 2000. The number of bits per variable is 8.Thus, if a rule has the n parameters, the total number of bits inthe string describing each member of the solutions is (n 8).ions 47 (2008) 3244Fig. 2. System modeling scheme.Aspiration criterion: TS has another important element calledaspiration mechanism. If a move on the tabu list leads to asolution with a value strictly better than the best obtained sofar, it is possible to allow this move to get out of the tabu list.This property is used to avoid removing very good moves fromconsideration and plays an essential role in the search process.In this work, if all available moves are classified as tabu, then aleast tabu move is selected for a new solution. The least tabumove means that this solution is changed less recently andfrequently among them but it is still classified as tabu. In thiswork, maximum number of iterations reached is selected as thestopping criterion of the search process.The basic procedure of TSA to be utilized in this paper isdescribed as follows:Initial solution;While predetermined stopping criteria not satisfied;{Create a set of neighbour solutions;Evaluate the neighbour solutions;Choose the best admissible solution;Test the tabu conditions;Perform the aspiration test;Update tabu list;}This procedure is repeated until a satisfactory solution isobtained or predetermined number of generations is elapsed.For all of the modeling examples, the number of the generationwas selected as 2000.3. Structure of the identified fuzzy rule baseFig. 2 shows the system modeling scheme. In this figure,yp(t + 1) is the output of the system to be modeled, andym(t + 1) is the output of the fuzzy model optimized by theTSA. Input of the TSA is the error value, and it is calculatedby {error = yp(t + 1) ym(t + 1)}. The major objective ofthe algorithm is to determine the most appropriate fuzzy modelparameters and to reduce the error factor.In fuzzy models, structure selection involves the followingchoices:(b) structure of the rulesionuzzin the fuzzy rule base are described by the membershipfunshaIntrapfuzvarandA rRiwhoutinnumAijInmeofshaissusysnofuzzy rule base as a probable solution element on which theureandputOnetontionThezzyatedtersrulerulethetrix.putberbers)]}.tabuachturetheareizedctions. Fuzzy membership functions can have differentpes depending on the designers preference or experience.general, fuzzy engineers have found that triangular andezoidal shapes help capture the modellers sense ofzy numbers and simplify computation. In this study, inputiables are characterized by triangular membership functions,the output variables are characterized by fuzzy singletons.ule in the rule base has a general form expressed by: If x1 is Ai1(x1) and x2 is Ai2(x2) then y is Bi (1)ere Ri is the i th rule, xj is the input variable, and y is theput variable. Aij(xj) is the fuzzy linguistic value defined asEq. (2) and Bi takes the role of fuzzy singleton, i.e., realber.= max(min[(xj a)/(b a), (c xj)/(c b)], 0). (2)Eq. (2); a, b, and c are the main parameters of a triangularmbership function. These parameters represent the locationsstarting point, peak point, and the ending point for a triangleped membership function, respectively.Deciding the number of fuzzy rules is a very criticale since it plays key roles in the fuzzy modeling of thetems. Except for some propositions, unfortunately, there isTSA operates. Selecting an appropriate encoding procedis a critical factor that affects the efficiency of the TSAthe final solution. The membership functions of the invariables are characterized by three numerical values.the other hand, the output variables are defined by singlnumerical values. These numerical values represent the posiof the membership function on the universe of discourse.parameter matrix defining the membership functions and furule base represented by the parameter matrices is formulin Fig. 3. The parameter matrix that contains the paramefor defining the membership functions within the fuzzybase will be of one-dimensional matrix form. For eachin the fuzzy rule base, the adjustment parameters used inreasoning procedure are also involved in this parameter maIn Fig. 3, these parameters are stated by di for each invariable. Thus, the size of matrix is determined by {the numof rules [(the number of input variables 3) + (the numof output variables 1) + (the number of input variableThis value of the matrix is also equal to the size of thelist. For a fuzzy model with two inputs and one output, efuzzy rule is represented by nine parameters. The rule struccoding is organized as shown in Fig. 3(c). For example, ifmodeling system has two inputs and one output, 144 bitsused to represent a probable solution with two rules optimA. Bagis / ISA TransactFig. 3. (a) Parameter matrix representing the membership functions, (b) F(c) number and type of membership functions for each variable(d) type of the inference mechanism, connective operators,defuzzification method.After the structure is fixed, the performance of a fuzzymodel can be fine-tuned by adjusting its parameters. Tunableparameters of linguistic models are the antecedent parametersand consequent membership functions and the rules. Rulessystematic and efficient procedure for choosing the mosts 47 (2008) 3244 35y rule base represented by parameter matrices, (c) The rule base coding.appropriate rule number. A reasonable number of fuzzy ruleswithout losing too much information about the system tobe modeled must be carefully obtained. In this work, usingdifferent rule sizes we have tried to achieve a good balancebetween the model performance and rule complexity.Another important element in combining a TSA and afuzzy model is the encoding method used to represent aby the algorithm (Fig. 3(c)). Initializing the solution is basedc1 N d 2MSE =Nk=1(Ok Ok) (5a)PI =Nk=1(Odk Ok)2Nk=1Odk (5b)where Odk , k = 1, . . . , N are the actual or desired output valuesand Ok , k = 1, . . . , N are the outputs from the model.These performance indices provide a means for evaluatingthe performance of a fuzzy model with the selected fuzzyrule base in the process of evolution, so that an optimizedmodel would be developed by the best individual. During themeasurements. The inputs to the fuzzy model were selected asu(t4), y(t1), respectively. The model output is y(t). For thevariables of u(t4), y(t1), and y(t), normalization intervalswere selected as [3, 3], [44, 62], and [44, 62], respectively.Parameters of the fuzzy model were searched into the intervalof [0, 4]. During the evolution, the modeling error of the bestindividual at each generation was calculated by Eq. (5a). Theoptimized fuzzy model for a gas furnace and the comparison ofmodel outputs and the original output data are shown in Figs. 4and 5, and the modeling error is 0.1483. The number of rulesin the fuzzy model is only four. Normalized and denormalizedparameter matrices of the fuzzy model are also given in Table 2.In Table 3, our fuzzy model is compared with other modelsidentified from the same data.36 A. Bagis / ISA TransaTable 2Optimized parameter matrix for fuzzy rule baseRules Input 1, u(t 4) Input 2, y(t ai1 ai2 ai3 bi1Normalized parameters(1) 0.1579 0.1720 0.4845 0.1485(2) 0.0552 0.3294 2.5020 0.0000(3) 0.0000 0.1569 0.8908 0.3476(4) 0.5640 0.7529 1.2549 0.2813Denormalized parameters(1) 2.0526 1.9680 0.0930 46.6730(2) 2.6688 1.0236 12.0120 44.0000(3) 3.0000 2.0586 2.3448 50.2568(4) 0.3840 1.5174 4.5294 49.0634on the randomly generated values from the given searchingintervals.In this study, the following reasoning procedures areconsidered:(a) Given input data x1 and x2, calculate the degree of thefulfillment i in the premise for the i th rule as ini = Ai1(x1k).di1 + Ai2(x2k).di2,k = input data number. (3)(b) Calculate the inferred value yi by taking the weightedaverage of Bi with respect to i as inyi =ri=1i.Biri=1i(4)where r is the number of fuzzy rules.To evaluate the performance of a fuzzy model, there are twoimportant performance indices in the literature. These are themean squared error (MSE) and a performance parameter statedas PI. In the MSE, the error is the difference between the actualoutput and the estimated output by the fuzzy model. The MSEand PI are calculated from N data point asevolution, the quality of the best solution at each generation iscalculated by Eq. (5a) or (5b).tions 47 (2008) 32441) Output, y(t) Adjustment parametersbi2 bi3 ci di1 di20.3595 1.0196 0.7658 0.4863 0.28130.2667 0.8046 0.0156 0.1489 0.50201.8823 2.7922 0.9412 0.4706 1.12140.4072 0.8314 0.2500 0.2510 0.078450.4710 62.3528 57.7844 0.4863 0.281348.8006 58.4828 44.2808 0.1489 0.502077.8814 94.2596 60.9416 0.4706 1.121451.3296 58.9652 48.5000 0.2510 0.07844. Simulation resultsFive numerical examples are provided in this section toillustrate the performance of the presented approach. All theseexamples have their origin in the existing literature. Thedatasets used in the first and third examples are derived frompractical processes; while the dataset used in the second andfourth examples is generated from the simulation of analyticalmodels. In the first example, the fuzzy modeling of a dynamicalprocess using a famous example of the system identificationgiven by Box and Jenkins [25] is discussed. The third exampleis used to show how to build a model of a human operatorscontrol action in a chemical plant. The structure of the humanoperation is shown in Sugeno and Yasukawa [6]. Furthermore,inputoutput data for human operation at the chemical processare also listed in [17]. In the fifth example, modeling of a realreservoir system is considered [3,4]. And, the datasets used inthis example are real flood hydrograph data of the dynamicalsystem.(a) Box and Jenkins gas furnace: The fuzzy model of adynamic process using a well-known example of the systemidentification given by Box and Jenkins is considered [6,22,25].The process is a gas furnace with a single input u(t) and a singleoutput y(t): gas flow rate and CO2 concentration, respectively.The dataset consists of N = 296 pairs of inputoutput(b) Modeling of a static nonlinear function: In this subsection,double input and single output static function was chosen astionruFig. 5. Comparison of modefurnace.a target system. The fuParthasarathy [13] and rey = (1+ x21 + x1.52 )2from which 50 data was50 inputoutput data areinputs x1 and x2 wereNormalization intervalswere selected as [0, 6]parameters were searcheevolution, the modelingby Eq. (5a). The optimiand the comparison of mdata are shown in Figs. 6error is 0.0019. The rWang and Langari (1996) [10] 5 0.1580.1750.0550.1110.1610.1530.0900.148arameter matrices of therformance indices of theods in the literature arey of the derived model,the model derivation areE calculated by (5a) isl plant: This exampleoperator control in aucing a polymer by thee start-up of the plant isused to manually operatestructure of the humanasukawa [6]. Five inputl outputs and original output data for the gasnction was taken from Narendra andpresented as, 1 x1, x2 5 (6)obtained. From this system equation,obtained. In this instance, two randominserted to test input identification.for inputoutput parameters x1, x2, y, [0, 6], [0, 10], respectively. Modeld in the interval of [0, 4]. During theerror at each generation was calculatedzed fuzzy model for nonlinear systemodel outputs and the original outputNakoula et al. (1997) [30] 90Kim et al. (1997) [8] 2Farag et al. (1998) [23] 37Kang et al. (2000) [22] 5Evsukoff et al. (2002) [16] 36Evsukoff et al. (2002) [16] 90Our model 4five. Normalized and denormalized pfuzzy model are given in Table 4. Pepresented method and the other methgiven in Table 5. To test the validitthe later 20 data that are not used inconsidered by the model, where MS0.0043 (Fig. 7(b)).(c) Human operation at a chemicadeals with the modeling of humanchemical plant. The plant is for prodpolymerization of monomers. Since thvery complicated, a human operator isthe system in this circumstance. Theoperation is given by Sugeno and YA. Bagis / ISA TransacFig. 4. Optimized fuzzyand 7, respectively, and the modelingule number of the fuzzy model iss 47 (2008) 3244 37le base for gas furnace.Table 3Comparison of our model with other modelsMethod Rule number MSEBox and Jenkins (1970) [25] 0.202Tong (1978) [26] 19 0.469Pedrycz (1984) [27] 81 0.320Takagi and Sugeno (1985) [5] 6 0.190Xu and Lu (1987) [28] 25 0.328Sugeno and Tanaka (1991) [29] 2 0.068Sugeno and Tanaka (1991) [29] 2 0.359candidates, u1: the monomer concentration, u2: the change ofmonomer concentration, u3: the monomer flow rate, u4 and u5:sactFig. 7. (a) Com for example (b).the local temperature inthe human operator maymonomer flow rate. Theabove six variables (fiveThere are different fuplant. In [6], a six ruleidentifying u1, u2, andand u3 were selected afrom the given data wepremise variables of thea fuzzy-neural system wperformance measuremes u1 and u3 were selectedization intervals for alls [4, 7], [0, 8000], andters were searched in theution, the modeling errorq. (5b). Optimized modelre given in Table 6 andase has only three rules.the original output dataparison of model outputs and original output data for example (b). (b) Test outputs and model outputsside the plant are available, to whichrefer. The output y is the set point forre are 70 data points for each of theinputs, and one output).zzy models available for modeling thisqualitative fuzzy model was given byu3 as input variables. In [10], u1, u2,s the input variables, and six clustersre taken to give six rules and threeTakagiSugenoKang model. In [15],In our approach, the input variableto build the fuzzy model. Normalvariables (u1, u3, y) were selected a[0, 8000], respectively. Model paramesolution area of [0, 2]. During the evolat each generation was calculated by Eparameters and optimum rule base aFig. 8, respectively. Optimum rule bThe comparison of model outputs andFig. 6. Optimized fuzzy rule base for a static nonlinear function.38 A. Bagis / ISA Tranas used for modeling the plant. Thent was not stated in [6] and [10].ions 47 (2008) 3244are shown in Fig. 9. In Table 7, our fuzzy model is comparedwith the other models in the literature.tioTable 5Comparison of different modeMethodSugeno and Yasukawa (1993)Lin and Cunningham (1995) [15] 3 0.0035Kim et al. (1997) [8] 3 0.0197Our model 5 0.0019(d) Nonlinear differential equation: This example is takenfrom Narendra and Parthasarathy [13] in which the plant tobe identified is given by the second-order highly nonlineardifference equationy(k) = y(k 1).y(k 2).(y(k 1)+ 2.5)1+ y2(k 1)+ y2(k 2) + u(k). (7)Training data of 500 points are generated from the plantmodel, assuming a random input signal u(k) uniformlydistributed in the interval [2, 2]. This unpredictable inputu(k) is randomly generated and is inserted into the system.Fig. 9. Comparison of model outputs and original output data for the chemicalplant.model, a different type sinusoidal input data u(k) is insertedinto the system y(k) (Fig. 10).The first 500 data poinidentification. To demoFig. 8. Optimized fuzzy rule base for human operation at a chemical plant.ls for example (b)Rule number MSE[6] 6 0.0790A. Bagis / ISA TransacTable 4Optimized parameter matrix for fuzzy rule baseRules Input 1, x1 Input 2, x2ai1 ai2 ai3 bi1Normalized parameters(1) 0.5348 0.7867 0.7887 0.3438(2) 0.2285 0.2286 0.5411 0.0000(3) 0.1661 0.2247 3.1662 0.1641(4) 0.1661 0.2516 0.2657 0.0938(5) 0.0938 0.0956 0.2657 0.2130Denormalized parameters(1) 3.2088 4.7202 4.7322 2.0628(2) 1.3710 1.3716 3.2466 0.0000(3) 0.9966 1.3482 18.9972 0.9846(4) 0.9966 1.5096 1.5942 0.5628(5) 0.5628 0.5736 1.5942 1.2780ts are produced from (7) for modelnstrate the performance of the fuzzyns 47 (2008) 3244 39Output, y Adjustment parametersbi2 bi3 ci di1 di20.7735 0.8282 0.1730 0.0625 1.49220.0728 0.3282 1.0000 0.1279 0.25000.3141 0.6661 0.2328 0.2731 1.04900.5781 1.9137 0.0313 0.3516 0.57420.3439 0.3594 0.9705 1.7887 0.05094.6410 4.9692 1.7300 0.0625 1.49220.4368 1.9692 10.000 0.1279 0.25001.8846 3.9966 2.3280 0.2731 1.04903.4686 11.4822 0.3130 0.3516 0.57422.0634 2.1564 9.7050 1.7887 0.0509The model has three inputs u(k), y(k 1), and y(k 2),and a single output y(k). Normalization intervals for the inputsTable 8Optimized parameter matrix for fuzzy rule baseRules Input 1, u(k) Input 2, y(k 1) Input 3, y(k 2) Output, y(k) Adjustment parametersai1 ai2 ai3 bi1 bi2 bi3 ci1 ci2 ci3 di ei1 ei2 ei3Normalized parameters(1) 0.0000 0.0001 0.8363 0.1212 0.4045 0.6214 0.0899 0.2462 1.0005 0.5226 1.3420 2.5905 1.2435(2) 0.0000 0.0000 0.3381 0.1250 0.1760 0.5904 0.0977 0.1055 1.9072 0.1055 0.7062 0.4260 1.4667(3) 0.3752 1.0179 1.9853 0.5814 0.9993 1.1009 0.9223 0.9231 0.9243 1.0924 1.3952 1.0081 0.4905(4) 0.1255 0.9849 3.8359 0.4385 0.8799 3.3353 0.5178 0.6566 0.9385 1.3333 0.4279 0.6614 0.3088Denormalized parameters(1) 2.000 1.999 1.345 3.788 0.955 1.214 4.101 2.538 5.005 0.2260 1.3420 2.5905 1.2435(2) 2.000 2.000 0.648 3.750 3.240 0.904 4.023 3.945 14.07 3.9450 0.7062 0.4260 1.4667(3) 0.499 2.072 5.941 0.814 4.993 6.009 4.223 4.231 4.243 5.9240 1.3952 1.0081 0.4905Table 7Comparison of different models for human operation at a chemical plantMethod ModelinputsRulenumberPISugeno and Yasukawa(1993) [6]u1, u2, u3 6 Not statedLin and Cunningham(1995) [15]u1, u2, u3 7 0.002245Wu and Yu (2000) [17] u3, u4 4 0.002000Our model u1, u3 3 0.001700were selected as [2, 2], [5, 5], and [5, 5], respectively.The search interval of all the variables is [0, 4]. After thelearning process is finished, the model is tested by applying asinusoidal input data u(k) = sin(2pik/25) to the fuzzy model.The optimized fuzzy model is given in Fig. 11. The rule numberof the fuzzy model is four. Normalized and denormalizedparameter matrices of the fuzzy model are also given in Table 8.The output of both the fuzzy model and the actual model areshown in Fig. 12. The fuzzy model has a good match withthe actual model with a MSE of 0.0378. Table 9 compares ourmodeling approach with the other approaches in the literature.The models are obtained from the previously generated 500data pairs and tested by applying a sinusoidal input signalu(k) = sin(2pik/25).(e) Modeling of a real reservoir system: This example includesthe modeling of the nonlinear relationship among the mainvariables of the reservoir in a reservoir management system.40 A. Bagis / ISA Transactions 47 (2008) 3244Table 6Optimized parameter matrix for fuzzy rule baseRules Input 1, u1 Input 2, u3 Output, y Adjustment parametersai1 ai2 ai3 bi1 bi2 bi3 ci di1 di2Normalized parameters(1) 0.0000 0.3130 1.1006 0.5938 1.4525 1.4642 0.8827 0.2544 1.4486(2) 0.1719 0.4840 0.7141 0.4527 0.4844 0.8908 0.4172 0.0774 0.3157(3) 0.2537 0.3387 0.6829 0.0430 0.0489 0.5266 0.0110 0.0351 0.8387Denormalized parameters(1) 4.0000 4.9390 7.3018 4750.40 11 620.00 11 713.60 7061.60 0.2544 1.4486(2) 4.5157 5.4520 6.1423 3621.60 3 875.20 7 126.40 3337.60 0.0774 0.3157(3) 4.7611 5.0161 6.0487 344.00 391.20 4 212.80 88.00 0.0351 0.8387(4) 1.498 1.940 13.344 0.615 3.799 28.35Fig. 10. Train and test inputs for a nonlinear differential equation.Table 9Comparison of different models for the nonlinear differential equationMethod RulenumberLearningMSETesting MSESugenos model (1991,1993) [6,29]12 0.5072 0.2447Farags model (1998) [23] 75 0.0374 0.0403Wangs model (1999) [7] 8 0.6184 0.2037Evsukoffs model (2002) [16] 100 0.1577 0.0185Our model 4 0.0341 0.03780.178 1.566 4.385 8.3330 0.4279 0.6614 0.3088tioforFig. 12. Comparisonnonlinear differential eReservoir managea spillway gate inof discharge wacomplex, nonlineaThe hydrologicalDetermination ofdifficult, (c) Consvery difficult, (d)control of the procThe main variam), spillway gate(Q, m3/s) [3,4,3nonlinear relationand spillway gateoutput variable of(Q). In this work,in Turkey is consitimized model parameters and optimum rule base are given inwas calculated fornd the results havebase has only fivethe original outputIn Fig. 15, processo dimensions. Themized by the TSAich has the struc-dation data modelt commonly usedthe time parameterThe values of thethe literature to beional performanceevaluation of theTable 12. In theand Pentium IVdeling based TSAear systems. Theulty in writing itsof model outputs and original output data for thequation.ment system is a control system that managesa dam to increase or decrease the amountter [3,4,31]. Reservoir management is ar, nonstationary control process because: (a)conditions have a nonlinear behavior, (b)the inflow hydrograph can be extremelytruction of a precise model of this process isOwing to human factor, accurate and efficientess is quite difficult.bles of a reservoir are reservoir lake level (H ,opening (d , m) and released water amount1]. The fuzzy model is used to model theship among these variables. Lake level (H )opening (d) are the inputs of the model. Thethe fuzzy model is the released water amountthe reservoir system of Adana Catalan DamTable 10 and Fig. 13, respectively. The PIthe model predictions of the validation data abeen given in Table 11. The optimum rulerules. The comparison of model outputs anddata is shown in Fig. 14 as three dimensions.outputs for train and test data are given as twresults show that the fuzzy rule structure optican yield a better model than the model whture of a neural network, improving the valiprediction PI.Because it did not seem to be the mosparameter in the literature to be considered,was not selected as a comparison parameter.computation time were not stated mostly incompared. However, to indicate the computatof the TSA, the computation time peralgorithm for each problem is given insimulations, Matlab programming package3600 MHz computer were used.It is clear from the figures that fuzzy mois a powerful tool for modeling of nonlinapproach is easy to use and there is no difficA. Bagis / ISA TransacFig. 11. Optimized fuzzy rule basedered, and the real flood hydrograph data forns 47 (2008) 3244 41the nonlinear differential equation.Catalan Dam are used [31]. To generate comparable results tothose already published; the 548 real data points were usedto generate a model using the TSA based method. At the endof the training process, the resultant model was applied to thevalidation dataset, which was the dataset of 25 unseen data. Allof the above information is obtained from the file containing thefinal project details of Catalan Dam [31].Normalization intervals for inputoutput parameters H, d, Qwere selected as [117, 131], [0, 12], [0, 10 000], respectively.Model parameters were searched in the interval of [0, 10]. Op-algorithm based modeling procedure.42TableOptimRule tersNorm(1) 610(2) 308(3) 965(4) 778(5) 429Deno(1) 117.79 117.93 125.82 0.0000 0.0000 6.4416 49.00 1.7488 2.4610(2) 135.67 0.0000 1.2308(3) 119.25 0.0293 1.1965(4) 123.56 0.2344 0.0778(5) 118.12 0.0510 1.1429Fig. 13. Optimized fuzzy rule base for a reservoir system.Fig. 14. Co173.00 177.45 6.4824 10.6644 33.8796 8105119.28 135.83 0.0708 6.9132 11.2512 5102125.99 129.64 9.0144 10.6860 11.9064 0.00120.42 123.62 0.0672 4.4532 10.1064 4859A. Bagis / ISA Transactions 47 (2008) 324410ized parameter matrix for fuzzy rule bases Input 1, H Input 2, d Output, Q Adjustment parameai1 ai2 ai3 bi1 bi2 bi3 ci di1 di2alized parameters0.0566 0.0661 0.6299 0.0000 0.0000 0.5368 0.0049 1.7488 2.41.3333 4.0000 4.3175 0.5402 0.8887 2.8233 0.8105 0.0000 1.20.1604 0.1631 1.3449 0.0059 0.5761 0.9376 0.5102 0.0293 1.10.4688 0.6421 0.9029 0.7512 0.8905 0.9922 0.0000 0.2344 0.00.0802 0.2444 0.4727 0.0056 0.3711 0.8422 0.4859 0.0510 1.1rmalized parametersmparison with three dimensions of the outputs for (a) train operation and (b) test operation in the reservoir system.(a)the beobviouof thegettingfor theotherntageshod isTable 12ComputProblem ation (s)BoxJeStatic nChemicNonlineequation)Reservost performance for all of the problem types considered, itsly exhibits a competitive performance for the modelingnonlinear or complex systems. It has the ability ofout of local minima and finding global optimal solutionsmodeling approach presented here is a powerful toolmodeling of nonlinear systems. As compared with themodeling methods, the described approach has the advaof simplicity, flexibility, and high accuracy. The metational performance of the TSA for different problemsDatanumberRulenumberParameter numberper ruleTotal parameternumberNeighbourhoodnumberComputation time per iternkins 296 4 9 36 288 9.18 (for 288 evaluation)onlinear function 50 5 9 45 360 2.37 (for 360 evaluation)al plant 70 3 9 27 216 1.27 (for 216 evaluation)ar differential 500 4 13 52 416 31.27 (for 416 evaluationFig. 15. Comparison with two dimensions of the outputs forTable 11Comparison of different models for a nonlinear reservoir systemMethod Model inputs PI (train) PI (test)Neural model [31] H, d 0.00170 0.0056Our model H, d 0.00158 0.0065When the model performance and rule size are considered,the resulting outcome is quite exciting. It is clear from the tablesthat a striking reduction in the value of the performance index(MSE or PI) is obtained without the need of a large number ofrules. The low rule size is a significant advantage of the model.The studies in the literature show that TSA is an effectiveoptimization method in the designing of fuzzy systems [3, 4].By using the control factors such as the initial solution, type ofmove, size of neighbourhood, tabu list size, aspiration criterionand stopping criterion, the performance of the algorithm can besignificantly improved. And, system models with high qualitycan be obtained.Three important conclusions can be made from the details ofthe numerical results:First, the requirement for using the large number of rules issuccessfully removed by employing the TSA. Tabu algorithmenables us to obtain better results when the number ofmembership functions and rules is as small as possible to solvethe problem at hand, a realistic case from an industrial point ofview. With tabu search, it is indeed possible to optimally tunethe structure of the fuzzy model.Second, even though the presented approach does not showir operation 548 5 9 4train operation and (b) test operation in the reservoir system.for multimodal problems in a reasonable time although thetraditional optimization algorithms fail to produce globaloptimal solution for such problems. Thus, it is verified thatthe presented method is capable of defining the nonlinearitiesor uncertainties of the system conditions without requiringany expert knowledge or experience about the system to beconsidered.Third, the numerical results show that the modelingapproach based on the TSA has high accuracy (low error)and good generalization ability. Moreover, it can start tolearn a process from a very low number of initial datasamples and improve the performance of the initial model.Therefore, it can be used effectively for various engineeringproblems in different areas with the nonlinear behaviors. And,accurate, fast, and reliable fuzzy models can be developed frommeasured/simulated data.When the process is highly nonlinear, time varying, or whenit depends on a lot of parameters, the cost for getting an accuratemodel is often estimated as being too high. In this case, fuzzymodeling approach based on the use of TSA can be viewed asa useful alternative method.5. ConclusionIn this paper, an efficient approach to fuzzy rule basedesign using the tabu search algorithm for nonlinear systemmodeling is presented. To demonstrate the effectiveness of thepresented approach, several numerical examples given in theliterature are examined. Simulation results show that the fuzzyA. Bagis / ISA Transactions 47 (2008) 3244 435 360 24.13 (for 360 evaluation)44 A. Bagis / ISA Transactions 47 (2008) 3244easy to use and there is no difficulty in writing its automaticmodeling procedure. Especially, when the model performanceand rule size are considered, the method based on the use of atabu search algorithm performs an important and very effectivemodeling procedure to fuzzy rule base design in the modelingof nonlinear or complex systems.AcknowledgementsThe author is grateful to the reviewers, and Prof. DervisKaraboga at Erciyes University in Turkey for their valuablecomments and suggestions.References[1] Babuska R. Fuzzy modeling for control. Kluwer Academic Pub.; 1998.[2] Nguyen HT, Sugeno M. Fuzzy systems: Modeling and control. KluwerAcademic Pub.; 1998.[3] Bagis A. Fuzzy and PD controller based intelligent control of spillway[15] Lin Y, Cunningham GA. A new approach to fuzzy-neural systemmodeling. IEEE Trans Fuzzy Syst 1995;3(2):1908.[16] Evsukoff A, Branco ACS, Galichet S. Structure identification andparameter optimization for non-linear fuzzy modeling. Fuzzy Sets andSystems 2002;132:17388.[17] Wu B, Yu X. 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ANFIS: adaptive network based fuzzy inference system. IEEETrans Syst Man Cybern 1993;23(3):66585.[13] Narendra KS, Parthasarathy K. Identification and control of dynamicalsystems using neural networks. IEEE Trans Neural Netw 1990;1(1):427.[14] Babuska R, Verbruggen H. Neuro-fuzzy methods for nonlinear systemidentification. Ann Rev Control 2003;27:7385.[23] Farag WA, Quintana VH, Torres GL. A genetic based neuro-fuzzyapproach for modeling and control of dynamical systems. IEEE TransNeural Netw 1998;9(5):75667.[24] Pham DT, Karaboga D. Intelligent optimisation techniques: Geneticalgorithms, tabu search, simulated annealing and neural networks.Springer-Verlag; 2000.[25] Box GEP, Jenkins GM. Time series analysis, forecasting, and control. SanFrancisco (CA): Holden Day; 1970.[26] Tong RM. Synthesis of fuzzy models for industrial processes-some recentresults. Int J General Syst 1978;(4):14362.[27] Pedrycz W. An identification algorithm in fuzzy relational systems. FuzzySets and Systems 1984;(13):15367.[28] Xu CW, Lu YZ. Fuzzy model identification and self-learning for dynamicsystems. IEEE Trans Syst Man Cybern 1987;(17):6839.[29] Sugeno M, Tanaka K. Successive identification of a fuzzy model and itsapplication to prediction of a complex system. Fuzzy Sets and Systems1991;(42):31534.[30] Nakoula Y, Galichet S, Foulloy L. A learning method for structure andparameter identification of fuzzy linguistic models. In: Hellendoorn H,Driankov D, editors. Selected approaches to fuzzy model identification.Berlin: Springer; 1997. p. 281319.[31] Bagis A, Karaboga D. Artificial neural networks and fuzzy logic basedcontrol of spillway gates of dams. Hydrol Process 2004;18:2485501.Fuzzy rule base design using tabu search algorithm for nonlinear system modelingIntroductionTabu search algorithm (TSA)Structure of the identified fuzzy rule baseSimulation resultsConclusionAcknowledgementsReferences

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